A hemisphere is placed on a spherical bubble of radius 1. Find the maximum height of the bubble tower.

I'm a little confused on how to go about solving such a problem, would appreciate any advice you guys might have.

Thanks.

Sep 22nd 2008, 02:48 PM

shawsend

1 Attachment(s)

Alright, here's a plot. Now, suppose:

. Can you then construct a function which relates the height of second hemisphere above the first (the orange line), as a function of the diameter of the second hemisphere (the green line)? Then take derivatives, set it equal to zero and get the maximum? I get

Sep 23rd 2008, 03:10 AM

Hweengee

the first figure is actually a sphere, but thanks for your help, i'll go work it out.

Sep 23rd 2008, 11:18 AM

Hweengee

Is the value of 1/sqrt(2) which you got the radius of the small hemisphere? If so the maximum height would be just 1+ sqrt(2)?

Sep 23rd 2008, 11:30 AM

Jameson

Quote:

Originally Posted by Hweengee

the first figure is actually a sphere, but thanks for your help, i'll go work it out.

He knows that. It's just easier to look at it this way. You only need one part of the sphere at a time.

Like shawsend said, you need to find a way to express a function that outputs the height of the bubble over 1 (the orange line).

This is the hardest part in these problems. Making weird substitutions so you can get a function of one variable and maximize it.

Sep 23rd 2008, 12:03 PM

shawsend

Hey, that triangle in there, isn't is just where is the radius of the small hemisphere (probably should have colored just half of the green line), b for blue, r for red?. Now, I want the blue line so that's . Now, look again at the figure. My ultimate goal is to express the height of the orange line as a function of the radius of the small hemisphere. Wouldn't that just be: . Alright then. Now I have the height of the second hemisphere above the first as a function of the radius of the second. How do I find the maximum height?

Sep 23rd 2008, 12:08 PM

Hweengee

i don't really get it, the height of the entire bubble tower would be blue line + r wouldn't it? so for an entire sphere it would be 1+r+sqrt(1-r^2)?

Sep 23rd 2008, 12:14 PM

shawsend

The entire height is blue+r. I'm just calculating the height of the bubble On the larger hemisphere (the orange line in the plot--click it to see a larger figure) and also I may have caused some confusion above. When I said I got , I should have said that is the maximum radius of the small hemisphere. The orange height at that value is actually which you can read off of the plot.

I don't understand what you mean by the entire sphere. My understanding was take a sphere of radius 1 then put a hollow hemisphere on top of it. My drawing is a cross-section of half of the sphere and the hollow hemisphere. What then is the maximum height of the hollow hemisphere above the sphere which I calculated is about .414.

Sep 23rd 2008, 12:15 PM

Hweengee

yea i got it now, so the maximum height can be obtained by placing a hemisphere of radius 1/sqrt(2) on the sphere. thanks.